A New Approach to Approximate Solutions for Nonlinear Differential Equation

In the last few years, the study of approximations methods for systems of differential equations has been extensively developed; see, for example, [1]. This technique, known as the perturbation method (see [2]), has many applications in the theory of fractional differentiation operators (see [3]), in reaction-diffusion equations, stochastic stability, and asymptotic stability (see [4–9]), and for some numerical considerations (see, for example, [10–12]). In current applications, some considerations require only the use of a small number of terms in the perturbation expansion, but the simple application of the perturbation is problematic if wewant to calculate a uniformly valid solution. Therefore, to structure a uniformly valid solution, one must look for an approximation that eliminates the terms causing the problem (secular terms). A technique to avoid the presence of these terms has been developed by Lindstedt. The principle of the Lindstedt method is to find approximations for periodic solutions, by convergent series using the expansion theorem and the periodicity of the solution [13, 14]. This method has various applications and properties; see, for example, [15]. Later, Poincaré proved that the expansion obtained by the Lindstedt technique is both asymptotic and uniformly valid. The aim of this work is to present an analytical approximation study of periodic solutions for systems of secondorder nonlinear differential equations. Although our analysis is based on the Lindstedt method, nevertheless the chosen development is according to a different approach from the one usually used. Thus, we recover an improvement in the process of the approximation. Our paper consists of three sections. In the first section we present the general framework of our study. In the second, we recall most of the preliminary notions and the necessary definitions, and we prove the third approximation in the general case. Finally, in Section 3 we define and study the approximations of a new nonclassical equation.

In current applications, some considerations require only the use of a small number of terms in the perturbation expansion, but the simple application of the perturbation is problematic if we want to calculate a uniformly valid solution.
Therefore, to structure a uniformly valid solution, one must look for an approximation that eliminates the terms causing the problem (secular terms).A technique to avoid the presence of these terms has been developed by Lindstedt.The principle of the Lindstedt method is to find approximations for periodic solutions, by convergent series using the expansion theorem and the periodicity of the solution [13,14].This method has various applications and properties; see, for example, [15].Later, Poincaré proved that the expansion obtained by the Lindstedt technique is both asymptotic and uniformly valid.
The aim of this work is to present an analytical approximation study of periodic solutions for systems of secondorder nonlinear differential equations.Although our analysis is based on the Lindstedt method, nevertheless the chosen development is according to a different approach from the one usually used.Thus, we recover an improvement in the process of the approximation.
Our paper consists of three sections.In the first section we present the general framework of our study.In the second, we recall most of the preliminary notions and the necessary definitions, and we prove the third approximation in the general case.Finally, in Section 3 we define and study the approximations of a new nonclassical equation.

Preliminaries and Definitions
In this section, we present an approximation method, based on the expansion of a solution of a differential equation in a series in a small parameter.It is used to construct uniformly valid periodic solutions to second-order nonlinear differential equations in the form with (0, ) = , (/)(0, )(0) = 0, where 0 <  << 1 means that the positive parameter  is small enough to be close to zero and  is supposed to be an analytical function of (, ) and (/)(, ).

International Journal of Mathematics and Mathematical Sciences
If  = 0 we obtain the following nonperturbed problem: Before we discuss our subject, we present some basic concepts concerning the perturbation theory.Then we introduce the Lindstedt method, which we use to determine uniformly valid solutions, in order to find a closer approximate solution for (2) ( * * is closer to  than  * means that | −  * * | < | −  * |).For further developments concerning the Lindstedt method see [16,17].

Approximation Technique.
We assume that the ( + 1)th approximate solution of (1) can be written as The general procedure of the simple approximation is to substitute (3) into (1), develop in powers of , and put all coefficients of the powers of  equal to zero.This gives a system of linear nonhomogeneous differential equations that we can solve recursively.But the simple approximation takes us on a problem, if we need to calculate an analytical approximations of periodic solutions of nonlinear differential equations in the form given by (1).We illustrate this type of difficulty in the following example.

Secular Terms.
The conservation of a finite numbers of terms on the right-side of expansion (5) determines a function that is not only nonperiodic, but also unbounded as  → +∞.Definition 1.Terms such as   cos() or   sin() where ,  ∈ N * ,  ∈ N are called secular terms.
These expressions appear because expansion (5) is not uniformly valid.The existence of such expressions destroys the periodicity of expansion (5) when only a finite number of terms is conserved.Therefore, to obtain a uniformly valid solution, we must look for an approximation that eliminates secular terms.A technique to avoid the presence of secular terms and allows for an approximation that is valid for all time has been developed by Lindstedt-Poincaré as described above in what follows.
which is required.
Remark 3.Although the calculation of  3 () is very long, usually in applications, the fourth approximation is among the high orders that are often useful.For this reason, we give its equation in the next proposition.
Remark 4. The Lindstedt method gives only periodic solutions.

Our Results
3.1.General Formula.Practically, for many considerations we are forced to use a small number of terms in the perturbation expansion.We note here that the second and third terms are determined by ( 11) and ( 12) in [17].In the following proposition, (13) which determines the fourth term is explicitly stated.
Proposition 5.The general formula of ( 13) is Proof.First, (9) gives such that International Journal of Mathematics and Mathematical Sciences 5 with On the other hand, in the third order we have and and also and when we substitute (31) into we get (25).

Main Result.
In this essential part of our work, we deal with some nonclassical equations, more general than (1), and also different from the equation studied in [2].We consider equations in the following form: with ỹ(0, ) = , ( ỹ/)(0, ) = 0, where  is a small positive parameter and  is supposed to be an analytical function of ỹ(, ) and  ỹ/(, ).
The aim of this study is to construct a new approach to (2), which gives a closer approximate solution of (2) more than an approximate solution of (1).The relations between an approximate solution of (32) and that of (1) are determined by the following lemma.
(3) When we apply the periodicity condition (22) to (12), we have On the other hand, according to (17) the solution ỹ2 ( θ, 0) of ( 11) is given by  2) more than the approximate solutions of ( 1).
Remark 10.We note here that, in the fractional case, the existence of a positive solution of (32) is studied in [18].Remark 11.Although the Lindstedt-Poincaré method gives uniformly valid asymptotic expansions for periodic solutions of weakly nonlinear oscillations, i.e., 0 <  <  1 , the technique does not work if the amplitude of the oscillation is a function of time (see [16,17]).